Math 101b: Algebra Ii Part C: Semisimplicity

We have one week to talk about semisimple rings and semisimple modules (Chapter XVII). A semisimple R-module is a finite direct sum of simple modules M = S1 ⊕ · · · ⊕ Sn and a semisimple ring is a ring R for which all f.g. modules are semisimple. The main reasons that I am choosing this particular topic in noncommutative algebra is for the study of representations of finite groups which we will do after the break. If G is a finite group then a representation of G over C is the same as a module over the group ring C[G] (also written CG). Once we have the basic definitions it will be very easy to see that C[G] is a semisimple ring. This makes the representation theory of finite groups elementary. From now on, all rings will be associative rings with 1 6= 0 (which may or may not be commutative) and R-module will be understood to be left R-modules.